# non-solvable finite group with a certain character degrees

Is there any non-solvable finite group $G$ such that ${\rm cd}(G) =\{1, 3, 4, 5, 6, 10 \}$, where ${\rm cd}(G)$ is the set of irreducible character degrees of $G$. In other words we have that a normal subgroup $M$ of $G$ such that $G/M \cong A_5$. Now, how do we can construct a non-solvable finite group $G$ such that ${\rm cd}(G) =\{ 1, 3, 4, 5, a, b\}$ where $6$ divides $a$, $10$ divides $b$, and $(5, a)=(3,b)=1$ ?

• I don't understand why the first sentence implies the second sentence – Yemon Choi Jul 22 '17 at 15:00
• The first sentence is a special case of the second. In other words, the relation between the degrees are important. – Guest Jul 22 '17 at 15:40
• It is not true that ${\rm cd}(A_5) = \{1,2,3,4,5,6,10\}$. So the sentence beginning "In other words" is not true. – Derek Holt Jul 22 '17 at 16:08
• But then doesn't $|G| = 1^{2} + 3^{2} + 4^{2} + 5^{2} + 6^{2} + 10^{2} = 187$ make that impossible, since all groups of order $187 = 11 \cdot 17$ are cyclic? – DavidLHarden Jul 22 '17 at 18:04
• @DavidLHarden this is the set of irreducible character degrees but my understanding is that it ignores multiplicities. – YCor Jul 22 '17 at 18:08